function [J grad] = nnCostFunction(nn_params, ...
                                   input_layer_size, ...
                                   hidden_layer_size, ...
                                   num_labels, ...
                                   X, y, lambda)
%NNCOSTFUNCTION Implements the neural network cost function for a two layer
%neural network which performs classification
%   [J grad] = NNCOSTFUNCTON(nn_params, hidden_layer_size, num_labels, ...
%   X, y, lambda) computes the cost and gradient of the neural network. The
%   parameters for the neural network are "unrolled" into the vector
%   nn_params and need to be converted back into the weight matrices. 
% 
%   The returned parameter grad should be a "unrolled" vector of the
%   partial derivatives of the neural network.
%

% Reshape nn_params back into the parameters Theta1 and Theta2, the weight matrices
% for our 2 layer neural network
Theta1 = reshape(nn_params(1:hidden_layer_size * (input_layer_size + 1)), ...
                 hidden_layer_size, (input_layer_size + 1));

Theta2 = reshape(nn_params((1 + (hidden_layer_size * (input_layer_size + 1))):end), ...
                 num_labels, (hidden_layer_size + 1));

% Setup some useful variables
m = size(X, 1); %numbers of training set
         
% You need to return the following variables correctly 
J = 0;
Theta1_grad = zeros(size(Theta1));
Theta2_grad = zeros(size(Theta2));

% ====================== YOUR CODE HERE ======================
% Instructions: You should complete the code by working through the
%               following parts.
%
% Part 1: Feedforward the neural network and return the cost in the
%         variable J. After implementing Part 1, you can verify that your
%         cost function computation is correct by verifying the cost
%         computed in ex4.m
%
%% Part 1;
X = [ones(m, 1), X]; % Add one columns to X; size=5000*1
yd = eye(num_labels);
y = yd(y, :); % trans y to number_label vector; size=5000 * 10

%% First to Hidden layer
a1 = X;
z2 = a1 * Theta1';
a2 = sigmoid(z2); % size=5000*25

%% Hidden to Output Layer
a2 = [ones(m, 1), a2];
z3 = a2 * Theta2';
a3 = sigmoid(z3); % size=5000*10

%% Compute Cost
% y has same size to a3, then using dot product to calu a 5000*10 matrix
logisf = (-y) .* log(a3) - (1-y) .* log(1-a3); 

%% Regularized cost
% dispite the first column that used to calu the 1 column
Theta1s = Theta1(:, 2:end); % size=25*400
Theta2s = Theta2(:, 2:end); % size=10*25
J = ((1 / m) .* sum(sum(logisf))) + (lambda / (2 * m)) .* (sum(sum(Theta1s .^ 2)) + sum(sum(Theta2s .^ 2)));
%%
% Part 2: Implement the backpropagation algorithm to compute the gradients
%         Theta1_grad and Theta2_grad. You should return the partial derivatives of
%         the cost function with respect to Theta1 and Theta2 in Theta1_grad and
%         Theta2_grad, respectively. After implementing Part 2, you can check
%         that your implementation is correct by running checkNNGradients
%
%         Note: The vector y passed into the function is a vector of labels
%               containing values from 1..K. You need to map this vector into a 
%               binary vector of 1's and 0's to be used with the neural network
%               cost function.
%
%         Hint: We recommend implementing backpropagation using a for-loop
%               over the training examples if you are implementing it for the 
%               first time.
%
%% set all delta D(bias error) to zeros
tridelta_1 = 0;
tridelta_2 = 0;
%% Compute delta, tridelta and D(BP Algorithm)
% output layer delta
delta_3 = a3 - y; %size=5000*10
% hidden layer delta
z2 = [ones(m, 1), z2]; %size=5000*26
delta_2 = delta_3 * Theta2 .* sigmoidGradient(z2); %(5000*10 * 10*26 .* 5000*26)=5000*26
delta_2 = delta_2(:, 2:end); %size=5000*25

tridelta_1 = tridelta_1 + delta_2' * a1; % same size as Theta1_grad (25*401)
tridelta_2 = tridelta_2 + delta_3' * a2; % same size as Theta2_grad (10*26)
Theta1_grad = (1 / m) .* tridelta_1;
Theta2_grad = (1 / m) .* tridelta_2;

%%
% Part 3: Implement regularization with the cost function and gradients.
%
%         Hint: You can implement this around the code for
%               backpropagation. That is, you can compute the gradients for
%               the regularization separately and then add them to Theta1_grad
%               and Theta2_grad from Part 2.
%
p1 = (lambda / m) .* [zeros(size(Theta1, 1), 1), Theta1(:, 2:end)];
p2 = (lambda / m) .* [zeros(size(Theta2, 1), 1), Theta2(:, 2:end)];
Theta1_grad = Theta1_grad + p1;
Theta2_grad = Theta2_grad + p2;










% -------------------------------------------------------------

% =========================================================================

% Unroll gradients
grad = [Theta1_grad(:) ; Theta2_grad(:)];


end
